Questions — Edexcel (10514 questions)

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Edexcel F2 2018 Specimen Q2
5 marks Standard +0.8
  1. Express \(\frac { 1 } { ( r + 6 ) ( r + 8 ) }\) in partial fractions.
  2. Hence show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 6 ) ( r + 8 ) } = \frac { n ( a n + b ) } { 56 ( n + 7 ) ( n + 8 ) }$$ where \(a\) and \(b\) are integers to be found.
    VIIN SIHI NI JIIIM IONOOVIIV SIHI NI III HM ION OOVI4V SIHI NI JIIIM IONOO
Edexcel F2 2018 Specimen Q3
10 marks Challenging +1.2
  1. Show that the substitution \(z = y ^ { - 2 }\) transforms the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 x y = x \mathrm { e } ^ { - x ^ { 2 } } y ^ { 3 }$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 4 x z = - 2 x \mathrm { e } ^ { - x ^ { 2 } }$$
  2. Solve differential equation (II) to find \(z\) as a function of \(x\).
  3. Hence find the general solution of differential equation (I), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
    VIIIV SIHI NI J14M 10N OCVIIN SIHI NI III HM ION OOVERV SIHI NI JIIIM ION OO
Edexcel F2 2018 Specimen Q4
9 marks Challenging +1.2
  1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { z - 1 } { z + 1 } , \quad z \neq - 1$$ The line in the \(z\)-plane with equation \(y = 2 x\) is mapped by \(T\) onto the curve \(C\) in the \(w\)-plane.
  1. Show that \(C\) is a circle and find its centre and radius. The region \(y < 2 x\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
  2. Sketch circle \(C\) on an Argand diagram and shade and label region \(R\).
    VIIIV SIHI NI IIIYM ION OCVIIV SIHI NI JIIIM ION OCVEXV SIHIL NI JIIIM ION OO
Edexcel F2 2018 Specimen Q5
9 marks Challenging +1.2
Given that \(y = \cot x\),
  1. show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 \cot x + 2 \cot ^ { 3 } x$$
  2. Hence show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = p \cot ^ { 4 } x + q \cot ^ { 2 } x + r$$ where \(p , q\) and \(r\) are integers to be found.
  3. Find the Taylor series expansion of \(\cot x\) in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\).
    VIIIV SIHI NI J14M 10N OCVIIN SIHI NI III HM ION OOVERV SIHI NI JIIIM ION OO
Edexcel F2 2018 Specimen Q6
13 marks Challenging +1.2
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = 2 \sin x$$ Given that \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 0\)
  2. find the particular solution of differential equation (I).
Edexcel F2 2018 Specimen Q7
8 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b197811e-1df5-4937-b0d8-f98f82412c76-24_480_926_217_511} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the two curves given by the polar equations $$\begin{array} { l l } r = \sqrt { 3 } \sin \theta , & 0 \leqslant \theta \leqslant \pi \\ r = 1 + \cos \theta , & 0 \leqslant \theta \leqslant \pi \end{array}$$
  1. Verify that the curves intersect at the point \(P\) with polar coordinates \(\left( \frac { 3 } { 2 } , \frac { \pi } { 3 } \right)\). The region \(R\), bounded by the two curves, is shown shaded in Figure 1.
  2. Use calculus to find the exact area of \(R\), giving your answer in the form \(a ( \pi - \sqrt { 3 } )\), where \(a\) is a constant to be found.
    VIIIV SIHI NI JIIIM ION OCVIIIV SIHI NI JIHM I I ON OCVI4V SIHI NI JIIYM IONOO
Edexcel F2 2018 Specimen Q8
14 marks Challenging +1.2
  1. Show that $$\left( z + \frac { 1 } { z } \right) ^ { 3 } \left( z - \frac { 1 } { z } \right) ^ { 3 } = z ^ { 6 } - \frac { 1 } { z ^ { 6 } } - k \left( z ^ { 2 } - \frac { 1 } { z ^ { 2 } } \right)$$ where \(k\) is a constant to be found. Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), where \(\theta\) is real,
  2. show that
    1. \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\)
    2. \(z ^ { n } - \frac { 1 } { z ^ { n } } = 2 i \sin n \theta\)
  3. Hence show that $$\cos ^ { 3 } \theta \sin ^ { 3 } \theta = \frac { 1 } { 32 } \quad ( 3 \sin 2 \theta - \sin 6 \theta )$$
  4. Find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 8 } } \cos ^ { 3 } \theta \sin ^ { 3 } \theta d \theta$$ \includegraphics[max width=\textwidth, alt={}, center]{b197811e-1df5-4937-b0d8-f98f82412c76-32_227_148_2524_1797}
Edexcel F2 Specimen Q4
10 marks Standard +0.3
4. $$z = - 8 + ( 8 \sqrt { } 3 ) \mathrm { i }$$
  1. Find the modulus of \(z\) and the argument of \(z\). Using de Moivre's theorem,
  2. find \(z ^ { 3 }\),
  3. find the values of \(w\) such that \(w ^ { 4 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
Edexcel F2 Specimen Q5
10 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd449136-cb09-49eb-8812-c863c0e7bd4e-10_506_728_267_632} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curves given by the polar equations $$r = 2 , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ and \(\quad r = 1.5 + \sin 3 \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\).
  1. Find the coordinates of the points where the curves intersect. The region \(S\), between the curves, for which \(r > 2\) and for which \(r < ( 1.5 + \sin 3 \theta )\), is shown shaded in Figure 1.
  2. Find, by integration, the area of the shaded region \(S\), giving your answer in the form \(a \pi + b \sqrt { 3 }\), where \(a\) and \(b\) are simplified fractions. $$\left[ \begin{array} { l l l } \text { Leave } \\ \text { blank } \\ \text { " } \\ \text { " } \end{array} & \\ \text { " } & \\ \text { " } & \\ \text { " } & \\ \text { " } & \\ \text { " } & \\ \text { " } & \\ \text { " } & \\ \text { " } & \end{array} \right.$$
Edexcel F2 Specimen Q6
10 marks Challenging +1.2
A complex number \(z\) is represented by the point \(P\) in the Argand diagram.
  1. Given that \(| z - 6 | = | z |\), sketch the locus of \(P\).
  2. Find the complex numbers \(z\) which satisfy both \(| z - 6 | = | z |\) and \(| z - 3 - 4 \mathrm { i } | = 5\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac { 30 } { z }\).
  3. Show that \(T\) maps \(| z - 6 | = | z |\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle.
Edexcel F2 Specimen Q7
12 marks Challenging +1.2
  1. Show that the transformation \(z = y ^ { \frac { 1 } { 2 } }\) transforms the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - 4 y \tan x = 2 y ^ { \frac { 1 } { 2 } }$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 2 z \tan x = 1$$
  2. Solve the differential equation (II) to find \(z\) as a function of \(x\).
  3. Hence obtain the general solution of the differential equation (I). $$\left[ \begin{array} { l } \text { Leave } \\ \text { blank } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \\ \text { " } \end{array} \right.$$
Edexcel F2 Specimen Q8
14 marks Challenging +1.3
  1. Find the value of \(\lambda\) for which \(y = \lambda x \sin 5 x\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$
  2. Using your answer to part (a), find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$ Given that at \(x = 0 , y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5\),
  3. find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( x )\).
  4. Sketch the curve with equation \(y = \mathrm { f } ( x )\) for \(0 \leqslant x \leqslant \pi\).
Edexcel FP3 Q1
6 marks Standard +0.8
  1. Find the exact values of x for which
$$4 \tanh ^ { 2 } x - 2 \operatorname { sech } ^ { 2 } x = 3 ,$$ giving your answers in the form \(\pm \ln \mathrm { a }\), where a is real.
Edexcel FP3 Q2
7 marks Challenging +1.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63249f82-4eab-47bc-aeae-3af8ec737b51-2_499_828_651_621} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve with equation \(y = 2 \cosh \left( \frac { 1 } { 2 } x \right)\). The points \(A\) and \(B\) lie on the curve and have \(x\)-coordinates \(- \ln 2\) and \(\ln 2\) respectively. The arc of the curve joining \(A\) and \(B\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the exact area of the curved surface area formed.
(Total 7 marks)
Edexcel FP3 Q3
8 marks Challenging +1.8
3. Using the substitution \(\mathrm { x } = \frac { 3 } { \sinh \theta }\), or otherwise, find the exact value of $$\int _ { 4 } ^ { 3 \sqrt { } 3 } \frac { 1 } { x \sqrt { } \left( x ^ { 2 } + 9 \right) } d x$$ giving your answer in the form a ln b , where a and b are rational numbers.
(Total 8 marks)
Edexcel FP3 Q4
9 marks Challenging +1.2
4. \(y = \arctan ( \sqrt { } x ) , \quad x > 0,0 < y < \frac { \pi } { 2 }\).
  1. Find the value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(\mathrm { x } = \frac { 1 } { 4 }\).
  2. Show that \(2 x ( 1 + x ) \frac { d ^ { 2 } y } { d x ^ { 2 } } + ( 1 + 3 x ) \frac { d y } { d x } = 0\).
Edexcel FP3 Q5
10 marks Challenging +1.8
5. $$\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { \mathrm { n } } x \mathrm { dx } , \mathrm { n } \geqslant 0$$
  1. Show that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\), for \(n \geqslant 2\)
  2. Using the result in part (a), find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \sin ^ { 5 } x \cos x d x$$
Edexcel FP3 Q6
11 marks Standard +0.8
Referred to a fixed origin O , the points \(\mathrm { P } , \mathrm { Q }\) and R have coordinates \(( \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) , ( - 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } )\) and \(( 3 \mathbf { j } - 5 \mathbf { k } )\) respectively. The plane \(\Pi _ { 1 }\) passes through \(\mathrm { P } , \mathrm { Q }\) and R . Find
  1. \(\overrightarrow { \mathrm { PQ } } \times \overrightarrow { \mathrm { QR } }\),
  2. a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r }\). ( \(\mathbf { i } + \mathbf { j } - \mathbf { k }\) ) \(= 6\). The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line I .
  3. Find a vector equation of I, giving your answer in the form ( \(\mathbf { r } - \mathbf { a }\) ) \(\times \mathbf { b } = \mathbf { 0 }\).
Edexcel FP3 Q7
12 marks Standard +0.3
7. \(\quad \mathbf { A } = \left( \begin{array} { c c c } 2 & \mathrm { k } & 0 \\ 1 & 1 & 0 \\ 0 & - 2 & 1 \end{array} \right)\), where k is a constant. Given that \(\left( \begin{array} { c } 9 \\ 3 \\ - 2 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\),
  1. show that \(\mathrm { k } = 6\),
  2. find the eigenvalues of \(\mathbf { A }\). A transformation \(\mathrm { T } : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { A }\).
    The point P has coordinates \(( \mathrm { t } - 2 , \mathrm { t } , 2 \mathrm { t } )\) where t is a parameter.
  3. Show that, for any value of \(t\), the transformation \(T\) maps \(P\) onto a point on the line with equation \(x - 4 y - 4 = 0\) (5)
Edexcel FP3 Q8
12 marks Challenging +1.8
8. The point \(\mathrm { P } ( 5 \sec \mathrm { u } , 3 \tan \mathrm { u } )\) lies on the hyperbola H with equation \(\frac { \mathrm { x } ^ { 2 } } { 25 } - \frac { \mathrm { y } ^ { 2 } } { 9 } = 1\). The tangent to \(H\) at \(P\) intersects the asymptote of \(H\) with equation \(y = \frac { 3 } { 5 } x\) at the point \(R\) and the asymptote with equation \(\mathrm { y } = - \frac { 3 } { 5 } \mathrm { x }\) at the point S .
  1. Use differentiation to show that an equation of the tangent to H at P is $$3 x = 5 y \sin u + 15 \cos u$$
  2. Prove that P is the mid-point of RS.
Edexcel C12 2015 January Q2
7 marks Moderate -0.3
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-03_473_654_233_603} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the graph of \(y = \frac { 12 } { \sqrt { \left( x ^ { 2 } - 2 \right) } } , x \geqslant 2\) The table below gives values of \(y\) rounded to 3 decimal places.
\(x\)25811
\(y\)8.4852.5021.5241.100
  1. Use the trapezium rule with all the values of \(y\) from the table to find an approximate value, to 2 decimal places, for $$\int _ { 2 } ^ { 11 } \frac { 12 } { \sqrt { \left( x ^ { 2 } - 2 \right) } } \mathrm { d } x$$
  2. Use your answer to part (a) to estimate a value for $$\int _ { 2 } ^ { 11 } \left( 1 + \frac { 6 } { \sqrt { \left( x ^ { 2 } - 2 \right) } } \right) d x$$
Edexcel C12 2017 January Q2
7 marks Moderate -0.8
A circle, with centre \(C\) and radius \(r\), has equation $$x ^ { 2 } + y ^ { 2 } - 8 x + 4 y - 12 = 0$$ Find
  1. the coordinates of \(C\),
  2. the exact value of \(r\). The circle cuts the \(y\)-axis at the points \(A\) and \(B\).
  3. Find the coordinates of the points \(A\) and \(B\).
Edexcel C12 2019 June Q3
6 marks Moderate -0.3
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de511cb3-35c7-4225-b459-a136b6304b78-06_955_1495_217_226} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve crosses the coordinate axes at the points \(( - 6,0 )\) and \(( 0,3 )\), has a stationary point at \(( - 3,9 )\) and has an asymptote with equation \(y = 1\) On separate diagrams, sketch the curve with equation
  1. \(y = - \mathrm { f } ( x )\)
  2. \(y = \mathrm { f } \left( \frac { 3 } { 2 } x \right)\) On each diagram, show clearly the coordinates of the points of intersection of the curve with the two coordinate axes, the coordinates of the stationary point, and the equation of the asymptote. \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-07_2255_45_316_36}
Edexcel C12 Specimen Q2
4 marks Easy -1.2
Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 3 - x ) ^ { 6 }$$ and simplify each term.
Edexcel C1 2005 January Q2
8 marks Easy -1.3
  1. Given that \(y = 5 x ^ { 3 } + 7 x + 3\), find
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), (b) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    2. Find \(\int \left( 1 + 3 \sqrt { } x - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\).